1547671870-The_Ricci_Flow__Chow

198 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

one concludes that

JV Rc\2 = \A\2 + \B\2 2 270 JV R\2'

whence Hamilton's estimate follows.

Combining Corollary 6.39 with Lemma 6.40 gives the following differen-

tial inequality.

COROLLARY 6.42. If (M^3 , g ( t)) is a solution of the Ricci flow on a

3-manifold, then

:t ( JRc\2 - tR2) :S ~ ( JRc\2 - tR2) - 327 \V Rc\2

26

• 8tr 9 (Rc^3 ) + ?;R 1Rcl^2 - 2R^3.

Now we return to equation (6.43) for the evolution of JV R\^2 / R. On

any manifold of positive Ricci curvature, one can estimate IRc\ :S R. Then

estimating

IV \Rc \

2

j :S 2 \V Re\ · \Re\

and recalling (6.46), we apply the Cauchy- Schwarz inequality to the bad

term~ (vR, V 1Rcl^2 ) on the right-hand side of (6.43), obtaining

(6.47)

~ (v R, V \Rc\

2

) ::; ~IV RI· IV \Rc\

2

1

:S 8 IV RI· JV Re\ \~\

::; 8J3 \V Rc\^2.

This motivates us to consider the quantity

(6.48) V ~ IV :12 + 327 ( 8v'3 + 1) ( JRcl2 - t R2)

which gives an upper bound for JV R\^2 / R. Combining Lemma 6.42 and

Corollary 6.42 with estimate (6.47) shows that V satisfies the following dif-

ferential inequality.

LEMMA 6.43. If (M^3 , g (t)) is a solution of the Ricci flow on a 3-

manifold whose Ricci curvature is positive initially, then

%t v::; ~v - 2R Jv (vRR) J

2

• 2l~c;2 1vR\

2

-1vRc1^2

3

2

7

( 8J3 + 1) (

2

3

6

R 1Rcl^2 - 8tr 9 (Rc^3 ) - 2R^3 ).